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Register a new userComputation of transmission eigenvalues by the regularized Schur complement for the boundary integral operators
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Added by
Yukun Guo
Publication date
2021
fields
Informatics Engineering
and research's language is
English
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This paper is devoted to the computation of transmission eigenvalues in the inverse acoustic scattering theory. This problem is first reformulated as a two by two boundary system of boundary integral equations. Next, utilizing the Schur complement technique, we develop a Schur complement operator with regularization to obtain a reduced system of boundary integral equations. The Nystr{o}m discretization is then used to obtain an eigenvalue problem for a matrix. We employ the recursive integral method for the numerical computation of the matrix eigenvalue. Numerical results show that the proposed method is efficient and reduces computational costs.
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In this paper, two types of Schur complement based preconditioners are studied for twofold and block tridiagonal saddle point problems. One is based on the nested (or recursive) Schur complement, the other is based on an additive type Schur complement after permuting the original saddle point systems. We discuss different preconditioners incorporating the exact Schur complements. It is shown that some of them will lead to positive stable preconditioned systems. Our theoretical analysis is instructive for devising various exact and inexact preconditioners, as well as iterative solvers for many twofold and block tridiagonal saddle point problems.
Boundary integral numerical methods are among the most accurate methods for interfacial Stokes flow, and are widely applied. They have the advantage that only the boundary of the domain must be discretized, which reduces the number of discretization points and allows the treatment of complicated interfaces. Despite their popularity, there is no analysis of the convergence of these methods for interfacial Stokes flow. In practice, the stability of discretizations of the boundary integral formulation can depend sensitively on details of the discretization and on the application of numerical filters. We present a convergence analysis of the boundary integral method for Stokes flow, focusing on a rather general method for computing the evolution of an elastic capsule, viscous drop, or inviscid bubble in 2D strain and shear flows. The analysis clarifies the role of numerical filters in practical computations.
The simulation of long, nonlinear dispersive waves in bounded domains usually requires the use of slip-wall boundary conditions. Boussinesq systems appearing in the literature are generally not well-posed when such boundary conditions are imposed, or if they are well-posed it is very cumbersome to implement the boundary conditions in numerical approximations. In the present paper a new Boussinesq system is proposed for the study of long waves of small amplitude in a basin when slip-wall boundary conditions are required. The new system is derived using asymptotic techniques under the assumption of small bathymetric variations, and a mathematical proof of well-posedness for the new system is developed. The new system is also solved numerically using a Galerkin finite-element method, where the boundary conditions are imposed with the help of Nitsches method. Convergence of the numerical method is analyzed, and precise error estimates are provided. The method is then implemented, and the convergence is verified using numerical experiments. Numerical simulations for solitary waves shoaling on a plane slope are also presented. The results are compared to experimental data, and excellent agreement is found.
We introduce a technique to automatically convert local boundary conditions into nonlocal volume constraints for nonlocal Poissons and peridynamic models. The proposed strategy is based on the approximation of nonlocal Dirichlet or Neumann data with a local solution obtained by using available boundary, local data. The corresponding nonlocal solution converges quadratically to the local solution as the nonlocal horizon vanishes, making the proposed technique asymptotically compatible. The proposed conversion method does not have any geometry or dimensionality constraints and its computational cost is negligible, compared to the numerical solution of the nonlocal equation. The consistency of the method and its quadratic convergence with respect to the horizon is illustrated by several two-dimensional numerical experiments conducted by meshfree discretization for both the Poissons problem and the linear peridynamic solid model.
We study the dependence of the continuity constants for the regularized Poincare and Bogovskiu{i} integral operators acting on differential forms defined on a domain $Omega$ of $mathbb{R}^n$. We, in particular, study the dependence of such constants on certain geometric characteristics of the domain when these operators are considered as mappings from (a subset of) $L^2(Omega,Lambda^ell)$ to $H^1(Omega,Lambda^{ell-1})$, $ell in {1, ldots, n}$. For domains $Omega$ that are star shaped with respect to a ball $B$ we study the dependence of the constants on the ratio $diam(Omega)/diam(B)$. A program on how to develop estimates for higher order Sobolev norms is presented. The results are extended to certain classes of unions of star shaped domains.
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